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Depending on your privacy settings, your newly added friend should be able to see your status updates yes.

Underneath every post in Facebook is a little icon, showing who can see that post. When you hover over it, it will tell you in more detail what the icon means. A globe means the post is Public (anyone can see it), 2 little persons mean only friends, the same 2 persons, but the right one in a lighter gray means friends except people who you added to the Acquaintances list, a padlock means only you, and a gear symbol means custom settings, which you can figure out by hovering over it. On your own posts, you can click this icon, and change it to whatever setting you want for that post. When adding a new post, you can also change it.

When a post has the privacy setting of Friends only, and you add somebody as a friend, they are now able to see that post. You could mark all your posts as Friends except Acquaintances, and add that person directly to your Acquaintances list, and they will not be able to see the post.

When you go to your own profile page, on the top right (just below your cover photo, next to the button Activity Log) is a gear icon. When you click that, and in the menu select View As, you can look at your own proflie through the eyes of somebody else. Like that you can see what that person could see on your profile.